Jtdylktuy

Does someone knows a good reference for the following result?

"A topological group is Hausdorff if and only if the identity is closed."

I have seen proofs in lecture notes of courses on the web, but I would like a reference in a book or an article, in order to refer to it.

Let $G$ be a topological group. By homogeneity, it is enough to prove that for every open set $V$ containing the neutral element $1$ there is an open set $U$ such that $$1in Usubseteq bar Usubseteq V.$$ Consider the map $mu:Gtimes Gto G$ defined by $$mu(g,h)= gcdot h^{-1}.$$ Since $G$ is a topological group, this map is continuous, and therefore $mu^{-1}(V)$ is open. It also contains the point $(1,1)$, so there is some open set $U$ containing $1$ such that $Utimes Usubseteq mu^{-1}(V)$ (basically by definiton of the product topology). This means that for every $u_1,u_2in U$ we have that $u_1cdot u_2^{-1}in V$ (and in particular $Usubseteq V$ since $1in U$). I claim that this $U$ is the one we are after; since we already have $1in Usubseteq V$, we just need to prove $bar Usubseteq V$.

Let $xin bar U$. The set $$xU={xcdot u|uin U}$$ is an open neighbourhood of $x$ and therefore intersects $U$. That is, there are $u_1,u_2in U$ such that $xcdot u_1 =u_2$. Therefore $x=u_1cdot u_2^{-1}in mu(Utimes U)subseteq V$, as needed.

This is probably the exact same proof as in Munkres only with a different choice of function as to avoid somewhat artificially restricting to symmetric neighbourhoods. To me it seems pretty intuitive.

The main ideas are that $mu$ is continuous and that if $W$ is an open set and $S$ is any set then $Scdot W$ contains in the closure of $S$.